Control and Synchronization of Neuron Ensembles Jr-Shin Li, Member, IEEE, Isuru Dasanayake, Student Member, IEEE, and Justin Ruths
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The Response of a Classical Hodgkin–Huxley Neuron to an Inhibitory Input Pulse
J Comput Neurosci (2010) 28:509–526 DOI 10.1007/s10827-010-0233-8 The response of a classical Hodgkin–Huxley neuron to an inhibitory input pulse Christoph Börgers · Martin Krupa · Stan Gielen Received: 23 June 2009 / Revised: 24 February 2010 / Accepted: 12 March 2010 / Published online: 13 April 2010 © The Author(s) 2010. This article is published with open access at Springerlink.com Abstract A population of uncoupled neurons can of- 1 Introduction ten be brought close to synchrony by a single strong inhibitory input pulse affecting all neurons equally. This Transient synchronization of neurons is believed to be mechanism is thought to underlie some brain rhythms, common in the brain and important for brain function. in particular gamma frequency (30–80 Hz) oscillations In a seminal paper, van Vreeswijk et al. (1994) showed in the hippocampus and neocortex. Here we show that that often synaptic inhibition, not excitation, leads to synchronization by an inhibitory input pulse often fails synchronized activity. Fast-spiking inhibitory interneu- for populations of classical Hodgkin–Huxley neurons. rons are believed to play the central role, in particular, Our reasoning suggests that in general, synchronization in the generation of gamma (30–90 Hz) rhythms (Traub by inhibitory input pulses can fail when the transition of et al. 1997; Whittington et al. 2000; Traub et al. 2003; the target neurons from rest to spiking involves a Hopf Csicsvari et al. 2003; Hájos et al. 2004;Mannetal.2005; bifurcation, especially when inhibition is shunting, not Compte et al. 2008). hyperpolarizing. Surprisingly, synchronization is more Arguably the simplest example of synchronizing in- likely to fail when the inhibitory pulse is stronger or hibition is that of a population of uncoupled neurons longer-lasting. -
Working Memory Cells' Behavior May Be Explained by Cross-Cortical Networks with Synaptic Facilitation
Working memory cells' behavior may be explained by cross-regional networks with synaptic facilitation S. Verduzco-Flores1, M. Bodner1,2,3, G.B. Ermentrout1, J. M. Fuster4, Y-D. Zhou2,3 1. University of Pittsburgh, Department of Mathematics, Pittsburgh PA 15102 2. MIND Research Institute, Santa Ana, CA 92704 3. Johns Hopkins University, Department of Neurosurgery, Baltimore, MD 21287 4. Semel Institute for Neuroscience and Human Behavior, University of California, Los Angeles, CA 90095 Corresponding Author: Mark Bodner Mind Research Institute 3631 South Harbor Blvd. Santa Ana, CA 92704 Phone: 714 751 5443 Fax: 714 751 5915 Email: [email protected] 1 2 Abstract Neurons in the cortex exhibit a number of patterns that correlate with working memory. Specifically, averaged across trials of working memory tasks, neurons exhibit different firing rate patterns during the delay of those tasks. These patterns include: 1) persistent fixed-frequency elevated rates above baseline, 2) elevated rates that decay throughout the tasks memory period, 3) rates that accelerate throughout the delay, and 4) patterns of inhibited firing (below baseline) analogous to each of the preceding excitatory patterns. Persistent elevated rate patterns are believed to be the neural correlate of working memory retention and preparation for execution of behavioral/motor responses as required in working memory tasks. Models have proposed that such activity corresponds to stable attractors in cortical neural networks with fixed synaptic weights. However, such networks typically do not reproduce the firing statistics or the range of patterned neuronal firing behaviors. For example, baseline and delay attractor states typically require sufficient separation (> 10 Hz) for stable memory. -
Supervised Learning in Spiking Neural Networks with FORCE Training
ARTICLE DOI: 10.1038/s41467-017-01827-3 OPEN Supervised learning in spiking neural networks with FORCE training Wilten Nicola1 & Claudia Clopath1 Populations of neurons display an extraordinary diversity in the behaviors they affect and display. Machine learning techniques have recently emerged that allow us to create networks of model neurons that display behaviors of similar complexity. Here we demonstrate the 1234567890 direct applicability of one such technique, the FORCE method, to spiking neural networks. We train these networks to mimic dynamical systems, classify inputs, and store discrete sequences that correspond to the notes of a song. Finally, we use FORCE training to create two biologically motivated model circuits. One is inspired by the zebra finch and successfully reproduces songbird singing. The second network is motivated by the hippocampus and is trained to store and replay a movie scene. FORCE trained networks reproduce behaviors comparable in complexity to their inspired circuits and yield information not easily obtainable with other techniques, such as behavioral responses to pharmacological manipulations and spike timing statistics. 1 Department of Bioengineering, Imperial College London, Royal School of Mines, London, SW7 2AZ, UK. Correspondence and requests for materials should be addressed to C.C. (email: [email protected]) NATURE COMMUNICATIONS | 8: 2208 | DOI: 10.1038/s41467-017-01827-3 | www.nature.com/naturecommunications 1 ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/s41467-017-01827-3 uman beings can naturally learn to perform a wide variety so on, humans use a very distinct tool to solve these problems: the Hof tasks quickly and efficiently. Examples include learning spiking neural network. -
Parabolic Bursting, Spike-Adding, Dips and Slices in a Minimal Model∗
Math. Model. Nat. Phenom. 14 (2019) 406 Mathematical Modelling of Natural Phenomena https://doi.org/10.1051/mmnp/2019018 www.mmnp-journal.org PARABOLIC BURSTING, SPIKE-ADDING, DIPS AND SLICES IN A MINIMAL MODEL∗ Mathieu Desroches1,2,∗∗, Jean-Pierre Francoise3 and Martin Krupa1,2,4 Abstract. A minimal system for parabolic bursting, whose associated slow flow is integrable, is pre- sented and studied both from the viewpoint of bifurcation theory of slow-fast systems, of the qualitative analysis of its phase portrait and of numerical simulations. We focus the analysis on the spike-adding phenomenon. After a reduction to a periodically forced one-dimensional system, we uncover the link with the dips and slices first discussed by J.E. Littlewood in his famous articles on the periodically forced van der Pol system. Mathematics Subject Classification. 37C10, 37C27, 34C23, 37G15, 34C15, 34E17. Received October 26, 2018. Accepted April 24, 2019. 1. Introduction Bursting is an essential rhythm of neural activity, whether it be intrinsic or extrinsic (upon the application of an external stimulus); ubiquitous in biological data, in particular from the brain, it can be modelled with mathematical equations within different frameworks, in particular parametrised ordinary differential equations (ODEs) with multiple timescale. Slow{fast models of bursting activity have been developed, both biophysically and phenomenologically, and analysed since the 1970s. The seminal work of Rinzel in the mid-1980s [39, 40] has lead to a first classification of bursting patterns using mathematical criteria, in particular bifurcation theory. Later on, this first classification was completed by Izhikevich [26] generalising the idea of using bifurcations of an auxiliary so-called fast subsystem to initiate and terminate the burst of a bursting system. -
Generalized Activity Equations for Spiking Neural Network Dynamics
HYPOTHESIS AND THEORY ARTICLE published: 15 November 2013 COMPUTATIONAL NEUROSCIENCE doi: 10.3389/fncom.2013.00162 Generalized activity equations for spiking neural network dynamics Michael A. Buice 1 and Carson C. Chow 2* 1 Modeling, Analysis and Theory Team, Allen Institute for Brain Science, Seattle, WA, USA 2 Laboratory of Biological Modeling, NIDDK, National Institutes of Health, Bethesda, MD, USA Edited by: Much progress has been made in uncovering the computational capabilities of spiking Julie Wall, Queen Mary, University neural networks. However, spiking neurons will always be more expensive to simulate of London, UK compared to rate neurons because of the inherent disparity in time scales—the spike Reviewed by: duration time is much shorter than the inter-spike time, which is much shorter than Brent Doiron, University of Pittsburgh, USA any learning time scale. In numerical analysis, this is a classic stiff problem. Spiking G. Bard Ermentrout, University of neurons are also much more difficult to study analytically. One possible approach to Pittsburgh, USA making spiking networks more tractable is to augment mean field activity models with *Correspondence: some information about spiking correlations. For example, such a generalized activity Carson C. Chow, Laboratory of model could carry information about spiking rates and correlations between spikes Biological Modeling, NIDDK, National Institutes of Health, self-consistently. Here, we will show how this can be accomplished by constructing a Building 12A Room 4007, 12 South complete formal probabilistic description of the network and then expanding around a Drive, Bethesda, MD 20814, USA small parameter such as the inverse of the number of neurons in the network. -
Supervised Learning in Spiking Neural Networks with FORCE Training
Supervised Learning in Spiking Neural Networks with FORCE Training Wilten Nicola1 and Claudia Clopath ∗1 1Department of Bioengineering, Imperial College London. Royal School of Mines, London UK, SW7 2AZ January 8, 2018 Acceptance Note that this manuscript has been accepted and published as of Dec 20th, 2017 in Nature Communications. Please cite the following when referencing this manuscript: Nicola, W., & Clopath, C. (2017). Supervised learning in spiking neural networks with FORCE training. Nature communications, 8(1), 2208. Abstract Populations of neurons display an extraordinary diversity in the behaviors they affect and display. Machine learning techniques have recently emerged that allow us to create networks of model neurons that display behaviours of similar complexity. Here, we demonstrate the direct applicability of one such technique, the FORCE method, to spiking neural networks. We train these networks to mimic dynamical systems, classify inputs, and store discrete sequences that correspond to the notes of a song. Finally, we use FORCE training to create two biologically motivated model circuits. One is inspired by the zebra-finch and successfully reproduces songbird singing. The second network is motivated by the hippocampus and is trained to store and replay a movie scene. FORCE trained networks reproduce behaviors comparable in complexity to their inspired circuits and yield information not easily obtainable with other techniques such as behavioral responses to pharmacological manipulations and spike timing statistics. Introduction Human beings can naturally learn to perform a wide variety of tasks quickly and efficiently. Examples include learning the complicated sequence of motions in order to take a slap-shot in Hockey or learning to replay the notes of a song after music class. -
The Function of Oscillations in the Hippocampal Formation 12
The Function of Oscillations in the Hippocampal Formation 12 Colin Lever, Raphael Kaplan, and Neil Burgess Abstract Some of the strongest experimental and computational links between oscillations and cognition concern the oscillations in the hippocampal formation supporting spatial and mnemonic processing. We review experimental and theoretical work concerning well-established hippocampal oscillations such as theta, gamma, and high-frequency ripples and how they relate to spatial, mnemonic, and anxiety- related representation and behaviour. We specifically consider the following computational roles for oscillations: organising processing into discrete chunks, as seen in encoding versus retrieval scheduling; ordinal and metric coding by oscillatory phase; temporal integration by oscillatory phase; and interregional communication. The literature on oscillations has typically been concerned with changes in band-specific power. Here, focusing on the theta oscillation, we summarise how key variables are linked not only to power but also to frequency and to coherence. We conclude that the hippocampal formation provides an invaluable model system for understanding the functional roles of neuronal oscillations and the interaction between oscillations. C. Lever ( *) Department of Psychology, University of Durham, Durham DH1 3LE, UK e-mail: [email protected] R. Kaplan Laboratory of Brain and Cognition, National Institute of Mental Health, Bethseda, MD, USA UCL Institute of Cognitive Neuroscience, London WC1N 3AR, UK N. Burgess ( *) UCL Institute of Cognitive Neuroscience, London WC1N 3AR, UK UCL Institute of Neurology, London WC1N 3BG, UK e-mail: [email protected] D. Derdikman and J.J. Knierim (eds.), Space, Time and Memory in the 303 Hippocampal Formation, DOI 10.1007/978-3-7091-1292-2_12, # Springer-Verlag Wien 2014 304 C. -
FAN-THESIS-2015.Pdf (1005.Kb)
BACKPROPGATION FOR CONTINUOUS THETA NEURON NETWORKS A Dissertation by DAVID FAN Submitted to the Office of Graduate and Professional Studies of Texas A&M University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Chair of Committee, Peng Li Committee Members, Yoonsuck Choe Arum Han Xiaoning Qian Head of Department, Miroslav M. Begovic May 2015 Major Subject: Electrical Engineering Copyright 2015 David Fan ABSTRACT The Theta neuron model is a spiking neuron model which, unlike traditional Leaky-Integrate-and-Fire neurons, can model spike latencies, threshold adaptation, bistability of resting and tonic firing states, and more. Previous work on learning rules for networks of theta neurons includes the derivation of a spike-timing based backpropagation algorithm for multilayer feedforward networks. However, this learn- ing rule is only applicable to a fixed number of spikes per neuron, and is unable to take into account the effects of synaptic dynamics. In this thesis a novel backpropa- gation learning rule for theta neuron networks is derived which incorporates synaptic dynamics, is applicable to changing numbers of spikes per neuron, and does not ex- plicitly depend on spike-timing. The learning rule is successfully applied to XOR, cosine and sinc function mappings, and comparisons between other learning rules for spiking neural networks are made. The algorithm achieves 97.8 percent training performance and 96.7 percent test performance on the Fischer-Iris dataset, which is comparable to other spiking neural network learning rules. The algorithm also achieves 99.0 percent training performance and 99.14 percent test performance on the Wisconsin Breast Cancer dataset, which is better than the compared spiking neural network learning rules. -
Review Theta Oscillations in the Hippocampus
Neuron, Vol. 33, 1–20, January 31, 2002, Copyright 2002 by Cell Press Theta Oscillations in the Hippocampus Review Gyo¨ rgy Buzsa´ ki1 1990; Vinogradova, 1995; Vertes and Kocsis, 1997). The Center for Molecular and Behavioral Neuroscience theta models discussed in previous reviews utilize neu- Rutgers rons with passive cable properties. During the past sev- The State University of New Jersey eral years, it has become increasingly clear that neurons and Neurological Institute of New Jersey are endowed with a host of active conductancies and Newark, New Jersey 07102 intrinsic oscillatory properties (Llina´ s, 1988; Ha¨ usser et al., 2000). A further goal of this review therefore is to integrate this new knowledge with network level infor- Theta oscillations represent the “on-line” state of the mation and propose new research directions for under- hippocampus. The extracellular currents underlying standing the functional significance of the theta rhythm. theta waves are generated mainly by the entorhinal input, CA3 (Schaffer) collaterals, and voltage-depen- Brain Systems Involved in the Generation ؉ dent Ca2 currents in pyramidal cell dendrites. The of Theta Oscillations rhythm is believed to be critical for temporal coding/ Theta oscillations depend on ongoing behavior (Gras- decoding of active neuronal ensembles and the modi- tya´ n et al., 1959; Vanderwolf, 1969). Although no consen- fication of synaptic weights. Nevertheless, numerous sus has yet emerged regarding their specific behavioral critical issues regarding both the generation of theta correlates, theta waves are most consistently present oscillations and their functional significance remain during REM sleep (Jouvet, 1969) and during various challenges for future research. -
Arxiv:1405.0044V2 [Q-Bio.NC] 8 Oct 2014 2
1 Correlations and functional connections in a population of grid cells Benjamin Dunn1;∗, Maria Mørreaunet1;∗, Yasser Roudi1;2;∗∗ 1 Kavli Institute for Systems Neuroscience and Centre for Neural Computation, NTNU, 7030 Trondheim, Norway 2 Nordita, KTH and Stockholm University, 16903 Stockholm, Sweden ∗ authors contributed equally ∗∗ E-mail: [email protected] Abstract We study the statistics of spike trains of simultaneously recorded grid cells in freely behaving rats. We evaluate pairwise correlations between these cells and, using a generalized linear model (kinetic Ising model), study their functional connectivity. Even when we account for the covariations in firing rates due to overlapping fields, both the pairwise correlations and functional connections decay as a function of the shortest distance between the vertices of the spatial firing pattern of pairs of grid cells, i.e. their phase difference. The functional connectivity takes positive values between cells with nearby phases and approaches zero or negative values for larger phase differences. We also find similar results when, in addition to correlations due to overlapping fields, we account for correlations due to theta oscillations and head directional inputs. The inferred connections between neurons can be both negative and positive regardless of whether the cells share common spatial firing characteristics, that is, whether they belong to the same modules, or not. The mean strength of these inferred connections is close to zero, but the strongest inferred connections are found between cells of the same module. Taken together, our results suggest that grid cells in the same module do indeed form a local network of interconnected neurons with a functional connectivity that supports a role for attractor dynamics in the generation of the grid pattern. -
A Model of Atropine-Resistant Theta Oscillations in Rat Hippocampal Area
Journal of Physiology (2002), 543.3, pp. 779–793 DOI: 10.1113/jphysiol.2002.024588 © The Physiological Society 2002 www.jphysiol.org A model of atropine-resistant theta oscillations in rat hippocampal area CA1 M. J. Gillies *, R. D. Traub ‡, F. E. N. LeBeau *, C. H. Davies †, T. Gloveli *, E. H. Buhl * and M. A. Whittington * * School of Biomedical Sciences, University of Leeds, Leeds LS2 9NQ, UK, † GlaxoSmithKline, Harlow, Essex, CM19 5AW, UK and ‡ Department of Physiology and Pharmacology, SUNY Health Science Center, Brooklyn, NY 11203, USA Theta frequency oscillations are a predominant feature of rhythmic activity in the hippocampus. We demonstrate that hippocampal area CA1 generates atropine-resistant theta population oscillations in response to metabotropic glutamate receptor activation under conditions of reduced Journal of Physiology AMPA receptor activation. This activity occurred in the absence of inputs from area CA3 and extra- ammonic areas. Field theta oscillations were co-expressed with pyramidal distal apical dendritic burst spiking and were temporally related to trains of IPSPs with slow kinetics. Pyramidal somatic responses showed theta oscillations consisted of compound inhibitory synaptic potentials with initial IPSPs with slow kinetics followed by trains of smaller, faster IPSPs. Pharmacological modulation of IPSPs altered the theta oscillation suggesting an inhibitory network origin. Somatic IPSPs, dendritic burst firing and stratum pyramidale interneuron activity were all temporally correlated with spiking in stratum oriens interneurons demonstrating intrinsic theta-frequency oscillations. Disruption of spiking in these interneurons was accompanied by a loss of both field theta and theta frequency IPSP trains. We suggest that population theta oscillations can be generated as a consequence of intrinsic theta frequency spiking activity in a subset of stratum oriens interneurons controlling electrogenesis in pyramidal cell apical dendrites. -
Lecture 1: Some Basics
Lecture 1: Some basics Habib Ammari Department of Mathematics, ETH Z¨urich Numerical methods for ODEs Habib Ammari Some basics • What is a differential equation? • Some methods of resolution: • Separation of variables; • Change of variables; • Method of integrating factors. • Important examples of ODEs: • Autonomous ODEs; • Exact equations; • Hamiltonian systems. Numerical methods for ODEs Habib Ammari Some basics • Ordinary differential equation (ODE): equation that contains one or more derivatives of an unknown function x(t). • Equation may also contain x itself and constants. • ODE of order n if the n-th derivative of the unknown function is the highest order derivative in the equation. Numerical methods for ODEs Habib Ammari Some basics • Examples of ODEs: • Membrane equation as a neuron model: dx(t) C + gx(t) = f (t); dt x(t): membrane potential, i.e., the voltage difference between the inside and the outside of the neuron; f (t): current flow due to excitation; C: capacitance; g: conductance (the inverse of the resistance) of the membrane. • Linear ODE of order 1. Numerical methods for ODEs Habib Ammari Some basics • Theta model: one-dimensional model for the spiking of a neuron. dθ(t) = 1 − cos θ(t) + (1 + cos θ(t))f (t); dt f (t): inputs to the model. • θ 2 [0; 2π]; θ = π the neuron spikes ! produces an action potential. • Change of variables, x(t) = tan(θ(t)=2), ! quadratic model dx(t) (∗) = x 2(t) + f (t): dt • Population growth under competition for resources: dx(t) r (∗∗) = rx(t) − x 2(t); dt k r and k: positive parameters; x(t): number of cells at time instant t, rx(t): growth rate and −(r=k)x 2(t): death rate.